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Students’ insight proves that the local-global conjecture doesn’t hold

quantamagazine.org

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Re: Students’ insight proves that the local-global conjecture doesn’t hold

#101

Post by commonlisp94 »

vkazanov wrote: post #94
commonlisp94 wrote: post #90

Earlier quoted context omitted.

> I will never need it as numeric methods took over completely What if you want to make a numerical method for something you can't look up the recipe for?

I don't think the point is that nobody needs to understand the theory. The point is that 3-5 years of analysis might be an overkill in most of practical situations. Numerical methods are quite well-studied by now. If you need a new method or a variation then you probably specialise in these things, and that's a different question.

It's really not. No problem looks exactly like the textbook you have to use analytical knowledge to devise a solution.

> The point is that 3-5 years of analysis might be an overkill

Rarely have I met an engineer and said, wow that guy knows TOO much about math. What do you think they should substitute?

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#102

Post by commonlisp94 »

vkazanov wrote: post #95
commonlisp94 wrote: post #91

Earlier quoted context omitted.

> Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations. And they use poor and imprecise approximations that they can't debug, because they just plug in numbers and hope for answer.

Understanding that something is fundamentally an approximation is important. Users of these packages are well aware of the fact. What point are you trying to make?

> fundamentally an approximation is important.

Yep, what you want is a good and predictable approximation.

> What point are you trying to make?

In practice most Matlab solutions you are describing are awful, in terms of performance and in terms of accuracy due to ignorance about underlying theory.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#103

Post by vkazanov »

commonlisp94 wrote: post #101
vkazanov wrote: post #94

Earlier quoted context omitted.

I don't think the point is that nobody needs to understand the theory. The point is that 3-5 years of analysis might be an overkill in most of practical situations. Numerical methods are quite well-studied by now. If you need a new method or a variation then you probably specialise in these things, and that's a different question.

It's really not. No problem looks exactly like the textbook you have to use analytical knowledge to devise a solution. > The point is that 3-5 years of analysis might be an overkill Rarely have I met an engineer and said, wow that guy knows TOO much about math. What do you think they should substitute?

I didn't say "too much math" :-) if anything, I'd like MORE math.

But the amount of time available is fundamentally limited and it has to be balanced.

E.g., in my personal circumstances probability theory, statistics, deeper proof understanding would all be useful. I had all of these at varying depths.

But it's only analysis that we were getting for years and years and years...

And I see why and how this is such an important subject historically. It's just not that big anymore relative to other things.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#104

Post by denton-scratch »

anon291 wrote: post #100
denton-scratch wrote: post #89

Earlier quoted context omitted.

Real numbers != Natural numbers. Every natural number has a 'next'; there is no 'next' for a real number.

> Every natural number has a 'next' Again... an axiomatic statement that is not well defined. What does it mean for something to have a 'next'. Shouldn't it be the case that if something has a next, then it can be named, identified, and perhaps even written down in some manner? Yet, by the same axioms, there certainly exist natural numbers that we cannot write down simply because there are not enough atoms in the uni…

> So basically, we have a conundrum, we say something exists after some other, yet for sure such a thing cannot be identified in any meaningful way, and its existence is just some conjecture that can never be proven.

Not being able to write a number down doesn't make it's existence a matter of conjecture. You could say the same thing about Pi.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#106

Post by anon291 »

denton-scratch wrote: post #104
anon291 wrote: post #100

Earlier quoted context omitted.

> Every natural number has a 'next' Again... an axiomatic statement that is not well defined. What does it mean for something to have a 'next'. Shouldn't it be the case that if something has a next, then it can be named, identified, and perhaps even written down in some manner? Yet, by the same axioms, there certainly exist natural numbers that we cannot write down simply because there are not enough atoms in the uni…

> So basically, we have a conundrum, we say something exists after some other, yet for sure such a thing cannot be identified in any meaningful way, and its existence is just some conjecture that can never be proven. Not being able to write a number down doesn't make it's existence a matter of conjecture. You could say the same thing about Pi.

> Not being able to write a number down doesn't make it's existence a matter of conjecture. You could say the same thing about Pi.

Depending on your philosophy of mathematics, there is good reason to believe that some real numbers do not exist. In particular, pi is a computable number, but many reals are not. Thus, we end up with a place where we conjecture that certain things exist yet simultaneously say there's (1) no way to write it down and (2) moreover, there's no systematic way to describe it. Given that with pi, there are many programs that given an N, can compute pi to N many digits, I do think it's reasonable to say that pi can be identified. But, there are infinitely many numbers that cannot. In fact, the vast majority of real numbers that supposedly exist cannot be computed to any arbitrary precision with a turing machine. Thus, they cannot be identified.

The general term for this philosophical approach towards mathematics is mathematical nominalism. What I'm seeing in this thread though is an implicit assumption that nominalism is false, despite being unaware of this assumption. I believe these sorts of hidden biases are dangerous. While I don't necessarily subscribe to nominalism, I think it's worth consideration, and I do think it brings up several interesting questions that cannot simply be ignored because 'well I believe it exists'.

References: https://plato.stanford.edu/entries/nominalism-mathematics/

More interesting reading:

https://philosophy.stackexchange.com/questions/81414/if-most...

https://math.stackexchange.com/questions/4322297/in-what-sen...

Many automated theorem provers can only prove things by construction, thus computability is the requirement for 'existence' in these systems (calculus of constructions via Coq, LEAN, etc). In other words, they follow a constructivist approach to mathematics, which is rather interesting as such approaches require us to elide a lot of 'obvious' axioms we take for granted (such as the law of excluded middle). Several things shake out of this approach such as the conclusion that all functions are continuous. (https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...)

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#107

Post by anon291 »

roywiggins wrote: post #92
denton-scratch wrote: post #64

Earlier quoted context omitted.

> We assume, for example, that there is always a number following another number Assuming you're referring to natural numbers or integers, that's not an assumption: https://proofwiki.org/wiki/Natural_Numbers_are_Infinite

That proof assumes that "s(n)=n+1 is clearly an injection" - ie that every natural number has a unique successor, and uses it to prove something about the size of the set. But it doesn't prove there is "always a number following another number", it assumes it. imho it's part of the definition of what the set of natural numbers even is. Peano arithmetic just has S(n) always exists and is injective as axioms.

Correct. All descriptions of the natural numbers of which I'm aware require the assumption that one can take some object and apply some operation 'S' on that object to produce another object that is neither the original object nor any of the objects produced before by repeated applications of 'S' to some root object. Whether you simply define a number as 0 and S(x) where x is a natural number or you define natural numbers as nested sets in ZFC or as church-encoded lambda terms, etc, you assume the existence of such a transformation.
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